And at the very end of the page, I want a save button, that can somehow grab all the values the user inputs in to all the forms, and pushes all this data to an array in a Service component. I'm not sure exactly how to do this part. Note that I don't want individual submit buttons for each dynamically generated form component.

You must import FormBuilder and FormGroup from @angular/forms like this:

import { FormBuilder, FormGroup } from '@angular/forms';

and then define in constructor:

constructor(private formBuilder: FormBuilder) { }

Step 2. Define new empty FormGrooup

Now you can define new empty FormGroup in ngOnInit like this:

ngOnInit() {
this.myForm = this.formBuilder.group({});
}

Step 3. Create FormControls dynamically (app.component.ts)

Now you can start with dynamically creation of your FormControls by iteration of classFields. For this I would recommend to create own function. This function gets two parameter: arrayName and classFields. With arrayName we can set custom name of our FormArray-control. classFields-Array we will use for iteration. We create constant variable for empty FormArray, which we called arrayControls. After this we iterate over classFields and create for each element FormControl, which we called control, and push this control into arrayControls. At the end of this function we add our arrayControls to our Form with custom name by using arrayName. Here is an example: